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We have an astronomical (Keplerian) telescope that we want to launch into space. First, however, we will try it on Earth, where we will measure the magnification $Z$. How does the distance between the lenses have to change for it to have the same magnification in space? Lenses have a refractive index of $n$.

1. How big must an aperture in a spatial filter be if we created it from a lens with a diameter of $40 \mathrm{cm}$ and its focal length is $4 \mathrm{m}$? Our Gaussian laser beam has an input diameter $30 \mathrm{cm}$ and a wavelength $1~053 nm$. The radius of the focus (parameter $\sigma $) of the Gaussian beam can be obtained using

\[\begin{equation*} r = \frac {2}{\pi }\lambda \frac {f}{D} \end {equation*}\] where $D$ is the diameter of the beam, $f$ is the focal length of the lens and $\lambda $ is the wavelength of the laser.

1. The laser beam is focused on a surface of a nuclear fuel pellet of a $1 \mathrm{mm}$ diameter. What energy should it have in order for the intensity in its focus to reach $10^{14} W.cm^{-2}$? The radius of the focus is $25 \mathrm{\micro m}$ and a pulse lasts $10 \mathrm{ns}$. How many beams do we need to equally cover the surface of a pellet? What is their total energy?
   What energy must the laser beam have if it is not focused on a surface of a nuclear fuel pellet, but the beam diameter matches exactly the diameter of the pellet and the density is its focus reaches $10^{14} W.cm^{-2}$? Assume that we have one such beam and it shines homogenously on the pellet „from all directions“.

Close to the shore, the speed of sea waves is influenced by the presence of the sea bed. Assume that the speed of waves $v$ is a function of the gravity of Earth $g$ and the water depth $h$. We have $v = C g^\alpha h^\beta $. Using dimensional analysis, determine the speed of the waves as a function of the depth. Constant $C$ is dimensionless, and cannot be determined using this method.

Besides the speed of the waves, swimming Jindra is also interested in the direction of incidence of the waves. Let's define a system of coordinates, where the water surface lies in the $xy$ plane. The shoreline follows the equation $y = 0$, the ocean lies in the $y > 0$ half-plane. The water depth $h$ is given as a function of distance from the shore $h = \gamma y$, where $\gamma = \const $. On the open ocean, where the speed of the waves is constant $c$ (not influenced by the depth), plane waves are propagating at incidence angle $\theta _0$ to the $x$ axis. Find a differential equation \[\begin{equation*} \der {y}{x} = \f {f}{y} \end {equation*}\] describing the shape of the wavefront close to the shore, but do not attempt to solve it, it is far from trivial. Calculate the incidence angle of the wavefront at the shoreline.

Bonus: Solve the differential equation and find the shape of the wavefront close to the shore.

Jindra walks down a long, lit corridor. His eyes are at a height of $1,7 \mathrm{m}$ above the floor, the light on the ceiling is at a height of $3,4 \mathrm{m}$. Jindra is now at a distance of $10 \mathrm{m}$ (horizontally) from the nearest light and is approaching it at a speed of $3 \mathrm{km\cdot h^{-1}}$. He sees a reflection of the light on the polished floor. How fast is the reflection approaching Jindra at this point?

A long time ago in a galaxy far, far away, one civilisation decided to move its whole solar system. One of the possibilities was to build a „Dyson half-sphere“, i. e. a megastructure which would capture approximately half of the radiation output of the start and reflect it in a single direction. An ideal shape would therefore be a paraboloid of revolution. What would be the relation between the radiation output of the star, surface mass density of such a mirror and its distance from the star such that this distance is constant?

Cathy and Catherine are watching a ship which is sailing with a constant speed towards a port. Cathy is standing on a rock above the port and her eyes are $h_1=20 \mathrm{m}$ above the surface of the water. Catherine is standing under the rock and her eyes are $h_2=1{,}7 \mathrm{m}$ above the surface of the water. If Catherine sees the top of the incoming ship $t=25 \mathrm{min}$ after Cathy sees it, what is the time of arrival of the ship to the port? Assume that the Earth is a perfect sphere with a radius $r=6378 \mathrm{km}$.

Find the refractive index of a transparent plane-parallel plate of thickness $d=1 \mathrm{cm}$, such that it will take one year for the light to pass through it. Discuss whether such a situation is possible.

Magic field of Discworld is so strong that the speed of light does no longer have its common meaning. This applies only close to the surface, where the refractive index of the magic field has magnitude $n_0 = 2,00 \cdot 10^{6}$. The refractive index decreases with height $h$ as $n(h) = n_0\eu ^{-kh}$, where $k = 1,00 \cdot 10^{-7} \mathrm{m^{-1}}$. Calculate the optimal angle (measured from vertical direction) under which a light signal shall be emitted from one end of the Discworld to reach the opposite end in the shortest time possible. Diameter of the Discworld is $d = 15\;000 \mathrm{km}$ and speed of light in vacuum is $c = 3,00 \cdot 10^{8} \mathrm{m\cdot s^{-1}}$.

The FYKOS bird found an optical bench at the Faculty of Physics. The bench allows him to place different tools along an optical axis. He started to play with it and gradually placed onto it: a point source of light, a first lens, a second lens and a screen, with the same spacing between them (so the distance between the screen and the light source is three times bigger than any distance of two neighbouring tools). A sharp image of the source was created on the screen. Then, he dipped the whole system into an unknown liquid, which he found in a strange container. To his amazement, the image on the screen stayed sharp. Figure out the refractive index of the given liquid, which is certainly different from the refractive index of air. You can assume that the refractive index of air is unitary. One of the lenses has ten times bigger focal length than the other and both are thin, manufactured from a material with refractive index $2$.

We illuminate a mirror at an angle of $\alpha = 15\mathrm{\dg }$ with respect to the normal. We want the light to travel directly back to the source. For doing so, we can use a glass prism with an index of refraction $n = 1,8$. Find the angle $\eta $ as a function of $\alpha $ and $n$ (see the figure). The prism is placed into the air with an index of refraction $n_0$.

Hint: \[\begin{align*} \sin \(x + y\) &= \sin x \cos y + \cos x \sin y  , \\
\cos \(x + y\) &= \cos x \cos y - \sin x \sin y  , \\
\sin x + \sin y &= 2\sin \(\frac {x + y}{2}\)\cos \(\frac {x - y}{2}\)  , \\
\cos x + \cos y &= 2\cos \(\frac {x + y}{2}\)\cos \(\frac {x - y}{2}\)  . \end {align*}\]